Consider the numbers n with at least two prime factors, the sum of whose prime factors divides n. In obvious analogy to the perfect numbers, I call these the

prime-perfect numbers. (Clearly, the sum of the prime factors of n is almost always less than n, so to require equality of n to the sum, as in the definition of perfect numbers, will be fruitless.)The sequence a(n) of prime-perfect numbers begins

30, 60, 70, 84, 90, 105, 120, 140, ....

(Note: This is EIS Sequence A066031.) The numbers k with just one prime factor have been excluded from the sequence since they trivially satisfy the requirement that the sum of the prime factors of k divide k. The exclusion thus highlights the interesting numbers satisfying the requirement.

It is easy to see that if p is a prime factor of the prime-perfect number n, then p

^{m}n is also prime-perfect for any m. Hence, a is an infinite sequence. But what about theelementary(orprimitive) terms of a, that is, terms which are not multiples of any previous terms? For example, 84 is elementary, since it is not a multiple of the preceding terms, 30, 60, 70. But 90 is not elementary because 90 is a multiple of 30. Are there also infinitely many elementary terms?A related problem: Find an expression generating elementary prime-perfect numbers.

I invite readers to communicate their solutions or comments by contacting me at the email address below. I will report any progress on these problems in this web page, and of course, acknowledge correct solutions.

Joseph L. Pe

iDEN System Engineering Tools and Statistics

Motorola Center

Schaumburg, ILemail:josephpe@excite.com

©2001 J. L. Pe. Document created on 12 December 2001 by J. L. Pe. Last updated on 13 December 2001 2001.

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